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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 5. Series and Product Developments}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}
\item Power Series Expansions 
\begin{enumerate}
\item[1.1.] Weierstrass's Theorem
\item[1.2.] The Taylor Series
\item[1.3.] The Laurent Series
\end{enumerate}

\item Partial Fractions and Factorization
\begin{enumerate}
\item[2.1] Partial Fractions
\item[2.2] Infinite Products
\item[2.3] Canonical Products
\item[2.4] The Gamma Function
\item[2.5] Stirling's Formula
\end{enumerate}

\item Entire Functions
\begin{enumerate}
\item[3.1] Jensen's Formula
\item[3.2] Hadamard's Theorem
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 4-5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}
\item[4.] The Riemann Zeta Function
\begin{enumerate}
\item[4.1] The Product Development
\item[4.2] Extension of $\zeta(s)$ to the Whole Plane
\item[4.3] The Functional Equation
\item[4.4] The Zeros of the Zeta Function
\end{enumerate}

\item[5.] Normal Families
\begin{enumerate}
\item[5.1] Equicontinuity
\item[5.2] Normality and Compactness
\item[5.3] Arzela's Theorem
\item[5.4] Families of Analytic Functions
\item[5.5] The Classical Definition
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Theorem 1. (Weierstrass) }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Suppose that $f_n(z)$ is analytic in the region $\Omega_n$, and that the sequence $\{f_n(z)\}$ converges to a limit function $f(z)$ in a region $\Omega$, uniformly on
every compact subset of $\Omega$.
Then $f(z)$ is analytic in $\Omega$. 
Moreover, $f_n'(z)$ converges uniformly to $f'(z)$ on every compact subset of $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
\item 
\item As a very simple example take $f_n(z) = z/(2z^n+1)$ and let $\Omega_n$ be the disk $|z| < 2^{-1/n}$. It is practically evident that $\lim f_n(z) = z$ in the disk
$|z| < 1$ which we choose as our region $\Omega$.



\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass Theorem - variant.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If a series with analytic terms,
$$
f(z) = f_1(z) + f_2(z) + \cdots + f_n(z) + \cdots,
$$
converges uniformly on every compact subset of a region $\Omega$, then the sum $f(z)$ is analytic in $\Omega$, and the series can be differentiated term by term. 

}

\item  Answer. 
\begin{enumerate}
\item 

\item 

\item 


\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Theorem 2. (A. Hurwitz)}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If the functions $f_n(z)$ are analytic and $\neq 0$ in a region $\Omega$,
and if $f_n(z)$ converges to $f(z)$, uniformly on every compact subset of $\Omega$, then $f(z)$ is either identically zero or never equal to zero in $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Exercise 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Using Taylor's theorem applied to a branch of $\log (1 + z/n)$,
prove that
$$
\lim\left(1 + \frac{z}{n}\right)^n = e^z
$$
uniformly on all compact sets.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Exercise 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the series
$$
\zeta(z) = \sum\limits_{n=1}^{\infty}n^{-s}
$$
converges for $\mathrm{Re} z > 1$, and represent its derivative in series form.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Exercise 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that
$$
(1-2^{1-z})\zeta(z) = 1^{-z} - 2^{-z} + 3^{-z} - \cdots
$$
and that the latter series represents an analytic function for $\mathrm{Re} z > 0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Exercise 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
As a generalization of Theorem 2, prove that if the $f_n(z)$ have at most $m$ zeros in $\Omega$, then $f(z)$ is either identically zero or has at most $m$ zeros.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.1. Weierstrass's Theorem. Exercise 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that
$$
\sum\limits_{n=1}^{\infty} \frac{nz^n}{1-z^n} 
= \sum\limits_{n=1}^{\infty}\frac{z^n}{(1-z^n)^2}
$$
for $|z| < 1$. (Develop in a double series and reverse the order of summation.)
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. The Taylor Series. Theorem 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $f(z)$ is analytic in the region $\Omega$, containing $z_0$, then the representation
$$
f(z) = f(z_0) + \frac{f'(z_0)}{1!}(z-z_0) + \cdots + \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n + \cdots
$$
is valid in the largest open disk of center $z_0$ contained in $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. The Taylor Series. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Develop $1/(1 + z^2)$ in powers of $z-a$, $a$ being a real number.
Find the general coefficient and for $a = 1$ reduce to simplest form.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. The Taylor Series. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The Legendre polynomials are defined as the coefficients $P_n(\alpha)$ in the development
$$
(1-2\alpha z + z^2)^{-\frac{1}{2}} = 1 + P_1(\alpha)z + P_2(\alpha)z^2 + \cdots.
$$
Find $P_1$, $P_2$, $P_3$, and $P_4$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{1.2. The Taylor Series. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Develop $\log (\sin z/z)$ in powers of $z$ up to the term $z^6$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{1.2. The Taylor Series. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What is the coefficient of $z^7$ in the Taylor development of $\tan z$?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{1.2. The Taylor Series. Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The Fibonacci numbers are defined by $c_0 = 0$, $c_1 = 1$,
$c_n = c_{n-1} + c_{n-2}$. 
Show that the $c_n$ are Taylor coefficients of a rational function, and determine a closed expression for $c_n$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that the Laurent development is unique.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Let $\Omega$ be a doubly connected region whose complement consists of the components $E_1$, $E_2$.
Prove that every analytic function $f(z)$ in $\Omega$ can be written in the form $f_1(z) + f_2(z)$ where $f_1(z)$ is analytic outside of $E_1$ and $f_2(z)$ is analytic outside of $E_2$.
(The precise proof requires a construction like the one in Chap. 4, Sec. 4.5.)
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The expression
$$
\{f,z\} = \frac{f'''(z)}{f'(z)} - \frac{3}{2}\left(\frac{f''(z)}{f'(z)}\right)^{2}
$$
is called the Schwarzian derivative of $f$. If $f$ has a multiple zero or pole,
find the leading term in the Laurent development of $\{f,z\}$.
Answer: If $f(z) = a(z-z_0)^m + \cdots$, then $\{f,z\} = \frac{1}{2}(1-m^2)(z-z_0)^{-2} + \cdots$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the Laurent development of $(e^z-1)^{-1}$ at the origin is of the form
$$
\frac{1}{z} - \frac{1}{2} + \sum\limits_{1}^{\infty}(-1)^{k-1}\frac{B_k}{(2k)!}z^{2k-1}
$$
where the numbers $B_k$ are known as the Bernoulli numbers. Calculate
$B_1$, $B_2$, $B_3$.
(By Sec. 2.1, Ex. 5, the $B_k$ are all positive.)
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. The Laurent Series. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Express the Taylor development of $\tan z$ and the Laurent development of $\cot z$ in terms of the Bernoulli numbers.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{2.1. Partial Fractions. Theorem 4.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Let $\{b_\nu\}$ be a sequence of complex numbers with $\lim\limits_{\nu\to\infty} b_\nu = \infty$, and let $P_\nu(\xi)$ be polynomials without constant term. 
Then there are functions which are meromorphic in the whole plane with poles at the points $b_\nu$ and the corresponding singular parts $P_\nu(1/(z-b_\nu))$. 
Moreover, the most general meromorphic function of this kind can be written in the form
$$
f(z)=\sum\limits_\nu \left[ P_\nu\left(\frac{1}{z-b_\nu}\right)-p_\nu(z) \right] +g(z). 
$$
where the $p_\nu(z)$ are suitably chosen polynomials and $g(z)$ is analytic in the
whole plane. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
%\item 

%\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Example - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\frac{\pi^2}{\sin^2\pi z} = \sum\limits_{n=-\infty}^{\infty} \frac{1}{(z-n)^2}
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Example - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\frac{\pi}{\sin\pi z} = \lim\limits_{m\to\infty}
\sum\limits_{n=-m}^{m} (-1)^n\frac{1}{z-n}
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Comparing coefficients in the Laurent developments of $\cot\pi z$ and of its expression as a sum of partial fractions, find the values of 
$$
\sum\limits_{1}^{\infty} \frac{1}{n^2}, \hspace{0.5cm}
\sum\limits_{1}^{\infty} \frac{1}{n^4}, \hspace{0.5cm}
\sum\limits_{1}^{\infty} \frac{1}{n^6}. 
$$
Give a complete justification of the steps that are needed.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Express
$$
\sum\limits_{-\infty}^{\infty} \frac{1}{z^3-n^3}
$$
in closed form.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Use 
$$
\frac{\pi}{\sin\pi z} = \lim\limits_{m\to\infty}
\sum\limits_{n=-m}^{m} (-1)^n\frac{1}{z-n}
$$
to find the partial fraction development of $1/\cos\pi z$, and show that it leads to 
$$
\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots .
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What is the value of
$$
\sum\limits_{-\infty}^{\infty} \frac{1}{(z+n)^2+a^2} ?
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. Partial Fractions. Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Using the same method as in Ex. 1, show that 
$$
\sum\limits_{-\infty}^{\infty} \frac{1}{n^{2k}}
 = 2^{2k-1}\frac{B_k}{(2k)!}\pi^{2k}. 
$$
(See Sec. 1.3, Ex. 4, for the definition of $B_k$.)
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{2.2. Infinite Products. Theorem 5.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The infinite product $\prod\limits_{1}^{\infty} (1 + a_n)$ with $1 + a_n \neq 0$  converges simultaneously with the series $\sum\limits_{1}^{\infty}\log (1 + a_n)$ whose terms represent the values of the principal branch of the logarithm.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Theorem 6. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A necessary and sufficient condition for the absolute convergence of the product 
$\prod\limits_{1}^{\infty} (1 + a_n)$ is the convergence of the series 
$\sum\limits_{1}^{\infty} |a_n|$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that 
$$
\prod\limits_{n=2}^{\infty} \left( 1-\frac{1}{n^2}\right) = \frac{1}{2}. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that for $|z| < 1$, 
$$
(1 + z)(1 + z^2)(1 + z^4)(1 + z^8) \cdots = \frac{1}{1-z}. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that
$$
\prod\limits_{1}^{\infty} \left( 1+\frac{z}{n}\right)e^{-z/n}
$$
converges absolutely and uniformly on every compact set.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that the value of an absolutely convergent product does not change if the factors are reordered.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Infinite Products. Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the function
$$
\theta(z) = \prod\limits_{1}^{\infty}(1+h^{2n-1}e^z)(1+h^{2n-1}e^{-z})
$$
where $|h| < 1$ is analytic in the whole plane and satisfies the functional
equation
$$
\theta(z+2\log h) = h^{-1} e^{-z}\theta(z).
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{2.3. Canonical Products. Theorem 7. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
There exists an entire function with arbitrarily prescribed zeros $a_n$ provided that, in the case of infinitely many zeros, $a_n\to\infty$. 
Every entire function with these and no other zeros can be written in the form
$$
f(z) = z^me^{g(z)}\prod\limits_{n=1}^{\infty}\left(1-\frac{z}{a_n}\right)
\exp\left[ \frac{z}{a_n} + \frac{1}{2}\left(\frac{z}{a_n}\right)^2+\cdots+\frac{1}{m_n}\left(\frac{z}{a_n}\right)^{m_n} \right]
$$
where the product is taken over all $a_n\neq 0$, the $m_n$ are certain integers, and
$g(z)$ is an entire function.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Canonical Products. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Suppose that $a_n\to\infty$ and that the $A_n$ are arbitrary complex numbers. 
Show that there exists an entire function $f(z)$ which satisfies 
$$f(a_n) = A_n.$$
}


\item  Answer. 
\begin{enumerate}
\item  Hint: Let $g(z)$ be a function with simple zeros at the $a_n$.
Show that
$$
\sum\limits_{1}^{\infty} g(z) \frac{e^{\gamma_n(z-a_n)}}{z-a_n}
\cdot \frac{A_n}{g'(a_n)}
$$
converges for some choice of the numbers $\gamma_n$.

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Canonical Products. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that
$$
\sin \pi(z + \alpha) = \exp(\pi z\cot\pi\alpha) 
\prod\limits_{-\infty}^{\infty}
\left(1+ \frac{1}{n+\alpha}\right)e^{-z/(n+\alpha)}
$$
whenever $\alpha$ is not an integer.
Hint: Denote the factor in front of the canonical product by $g(z)$ and determine $g'(z)/g(z)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Canonical Products. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What is the genus of $\cos \sqrt{z}$?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Canonical Products. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $f(z)$ is of genus $h$, how large and how small can the genus of $f(z^2)$ be?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. Canonical Products. Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that if $f(z)$ is of genus 0 or 1 with real zeros, and if $f(z)$ is real for real $z$, then all zeros of $f'(z)$ are real. 
Hint: Consider $\mathrm{Im} f'(z)/f(z)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. The Gamma Function. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. The Gamma Function. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove the formula of Gauss:
$$
(2\pi)^{\frac{n-1}{2}}\Gamma(z) = n^{z-\frac{1}{2}}
\Gamma\left(\frac{z}{n}\right)
\Gamma\left(\frac{z+1}{n}\right)\cdots
\Gamma\left(\frac{z+n-1}{n}\right). 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. The Gamma Function. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that
$$
\Gamma\left(\frac{1}{6}\right)=2^{-\frac{1}{3}}
\left(\frac{3}{\pi}\right)^{\frac{1}{2}}
\Gamma\left(\frac{1}{3}\right)^2.
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. The Gamma Function. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
What are the residues of $\Gamma(z)$ at the poles $z = -n$?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. Stirling's Formula. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
How is the Gamma function and the Stirling's formula related? 
}

\item  Answer. 
\begin{enumerate}
\item  In most connections where the $\Gamma$ function can be applied, it is of utmost importance to have some information on the behavior of $\Gamma(z)$ for very large values of $z$.

\item  Fortunately, it is possible to calculate $\Gamma(z)$ with great precision and very little effort by means of a classical formula which goes under the name of Stirling's formula. 

\item  There are many proofs of this formula. 

\item  We choose to derive it by use of the residue calculus, following mainly the presentation of Lindel\"{o}f in his classical book on the calculus of residues.

\item  This is a very simple and above all a very instructive proof inasmuch as it gives us an opportunity to use residues in less trivial cases than previously.

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. Stirling's Formula. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
\begin{equation*}
\begin{aligned}
\Gamma(z) &= \sqrt{2\pi} z^{z-\frac{1}{2}}e^{-z}e^{J(z)}, \\
J(z) &= \frac{1}{\pi}\int_0^{\infty} \frac{z}{\eta^2+z^2} \log \frac{1}{1-e^{-2\pi \eta}}d\eta.
\end{aligned}
\end{equation*}
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. Stirling's Formula. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove the development 
\begin{equation*}
\begin{aligned}
\Gamma(z) &= \sqrt{2\pi}z^{z-\frac{1}{2}}e^{-z}e^{J(z)}, \\
J(z) &= \frac{B_1}{1\cdot 2}\frac{1}{z} 
- \frac{B_2}{3\cdot 4}\frac{1}{z^3} 
+\cdots 
+ (-1)^{k-1} \frac{B_k}{(2k-1)\cdot 2k}\frac{1}{z^{2k-1}} + J_k(z).  
\end{aligned}
\end{equation*}
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. Stirling's Formula. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
For real $x > 0$ prove that
$$
\Gamma(x) = \sqrt{2\pi} x^{x-\frac{1}{2}}e^{-x}e^{\theta(x)/12x}
$$
with $0 < \theta(x) < 1$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.5. Stirling's Formula. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The formula 
$$
\Gamma(z) = \int_0^\infty e^{-t}t^{z-1}dt
$$
permits us to evaluate the probability integral
$$
\int_0^\infty e^{-t^2}dt = \frac{1}{2}\int_0^\infty e^{-x}x^{-\frac{1}{2}}dx 
= \frac{1}{2}\Gamma(\frac{1}{2}) = \frac{1}{2}\sqrt{\pi}. 
$$
Use this result together with Cauchy's theorem to compute the Fresnel integrals
$$
\int_0^\infty \sin (x^2)dx, \hspace{0.5cm}
\int_0^\infty \cos (x^2)dx.
$$
}

\item  Answer: Both are equal to $\frac{1}{2}\sqrt{\pi/2}$.

%\item  Answer. 
%\begin{enumerate}
%\item 
% 
%%\item 
%%
%%\item 
%
%\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.1. Jensen's Formula. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $f(z)$ is an analytic function, then $\log |f(z)|$ is harmonic except at the zeros of $f(z)$. Therefore, if $f(z)$ is analytic and free from zeros in $|z|\le\rho$, 
%(43)
$$
\log |f(0)| = \frac{1}{2\pi} \log |f(\rho e^{i\theta})| d\theta, 
$$
and $\log |f(z)|$ can be expressed by Poisson's formula.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.1. Jensen's Formula. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\log |f(0)| = -\sum\limits_{i=1}^{n}\log \left(\frac{\rho}{|a_i|} \right) + 
\frac{1}{2\pi} \int_0^{2\pi} \log |f(\rho e^{i\theta})| d\theta.
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.1. Jensen's Formula. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. Hadamard's Theorem. Theorem 8. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The genus and the order of an entire function satisfy the double inequality 
$h\le\lambda\le h+1$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. Hadamard's Theorem. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The characterization of the genus given in the first paragraph of Sec. 3.2 is not literally the same as the definition in Sec. 2.3. Supply the reasoning necessary to see that the conditions are equivalent.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. Hadamard's Theorem. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Assume that $f(z)$ has genus zero so that
$$
f(z) = z^m\prod\limits_{n} \left( 1-\frac{z}{a_n} \right). 
$$
Compare $f(z)$ with 
$$
g(z) = z^m\prod\limits_{n} \left( 1-\frac{z}{|a_n|} \right). 
$$
and show that the maximum modulus $\max_{|z|=r} |f(z)|$ is $\le$ the maximum modulus of $g$, and that the minimum modulus of $f$ is $\ge$ the minimum modulus of $g$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
%\item 

%\item 

\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{4.1. The Product Development. Theorem 9. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
For $\sigma=\mathrm{Re}s>1$, $$\frac{1}{\zeta(s)} = \prod\limits_{n=1}^\infty (1-p_n^{-s}). $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.2. Extension of $\zeta(s)$ to the Whole Plane. Theorem 10. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
For $\sigma>1$, $$\zeta(s) = -\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-z)^{s-1}}{e^z-1}dz$$
where $(-z)^{s-1}$ is defined on the complement of the positive real axis as 
$\exp[(s-1)\log(-z)]$ with $-\pi<\mathrm{Im} \log(-z)<\pi$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.3. The Functional Equation. Theorem 11. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$\zeta(s) = 2^s\pi^{s-1} \sin\frac{\pi s}{2} \Gamma(1-s)\zeta(1-s). $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{4.4. The Zeros of the Zeta Function. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item The famous Riemann conjecture, which has neither been proved nor disproved, asserts that all nontrivial zeros lie on the critical line $\sigma =1/2$. 

\item It is not too difficult to prove that there are no zeros on $\sigma = 1$ and $\sigma = 0$.

\item It is known that asymptotically more than one third of the zeros lie on the critical line.

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.1. Equicontinuity. Definition 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. What is an equicontinuous family of functions? }

\item  Answer. 
\begin{enumerate}
\item The functions in a family $\mathcal{F}$ are said to be equicontinuous on a set $E \subset \Omega$ if and only if, for each $\varepsilon > 0$, there exists a $\delta > 0$ such that $d(f(z),f(z_0)) < \varepsilon$ whenever $|z - z_0| < \delta$ and $z_0,z \in E$, simultaneously for all functions $f\in \mathcal{F}$. 

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.1. Equicontinuity. Definition 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. What is a normal family of functions?}

\item  Answer. 
\begin{enumerate}
\item  A family $\mathcal{F}$ is said to be normal in $\Omega$ if every sequence $\{f_n\}$ of functions $f_n \in\mathcal{F}$ contains a subsequence which converges uniformly on every compact subset of $\Omega$.
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}



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\begin{frame}{5.2. Normality and Compactness. Theorem 12.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A family $\mathcal{F}$ is normal if and only if its closure $\mathcal{F}^-$ with
respect to the distance function 
$$
\rho(f,g) = \sum\limits_{k=1}^{\infty} \frac{\sup_{z\in E_k} \delta(f(z),g(z))}{2^k}
$$
is compact. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. Normality and Compactness. Theorem 13.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The family $\mathcal{F}$ is totally bounded if and only if to every compact set $E\subset\Omega$ and every $\varepsilon > 0$ it is possible to find $f_1,\cdots,f_n\in\mathcal{F}$ such that every $f\in\mathcal{F}$ satisfies $d(f,f_j)<\varepsilon$ on $E$ for some $f_j$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Arzela's Theorem. Theorem 14. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A family $\mathcal{F}$ of continuous functions with values in a metric space $S$ is  normal in the region $\Omega$ of the complex plane if and only if 
\begin{enumerate}
\item  $\mathcal{F}$ is equicontinuous on every compact set $E\subset \Omega$;
\item  for any $z\in\Omega$ the values $f(z), f\in\mathcal{F}$ lie in a compact subset of $S$.
\end{enumerate}
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.4. Families of Analytic Functions. Theorem 15. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A family $\mathcal{F}$ of analytic functions is normal with respect to $\mathbb{C}$ if and only if the functions in $\mathcal{F}$ are uniformly bounded on every compact set.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.4. Families of Analytic Functions. Theorem 16.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A locally bounded family of analytic functions has locally bounded derivatives.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Definition 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item  A family of analytic functions in a region $\Omega$ is said to be normal if every sequence contains either a subsequence that converges uniformly on every compact set $E \subset \Omega$, or a subsequence that tends uniformly to $\infty$ on every compact set.
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Theorem 17. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A family of analytic or meromorphic functions $f$ is normal in the classical sense if and only if the expressions
$$
\rho(f) = \frac{2|f'(z)|}{1+|f(z)|^2}
$$
are locally bounded. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Prove that in any region $\Omega$ the family of analytic functions with positive real part is normal. Under what added condition is it locally bounded? Hint: Consider the functions $e^{-f}$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the functions $z^n$, $n$ a nonnegative integer, form a normal family in $|z| < 1$, also in $|z| > 1$, but not in any region that contains a point on the unit circle.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If $f(z)$ is analytic in the whole plane, show that the family formed by all functions $f(kz)$ with constant $k$ is normal in the annulus $r_1 < |z| < r_2$ 
if and only if $f$ is a polynomial.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.5. The Classical Definition. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
If the family $\mathcal{F}$ of analytic (or meromorphic) functions is not normal in $\Omega$, show that there exists a point $z_0$ such that $\mathcal{F}$ is not normal in any neighborhood of $z_0$. 
Hint: A compactness argument.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\end{document}

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